By Robert Harper

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Extra resources for Practical Foundations for Programming Languages 1.19

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2 Hypothetical Inductive Definitions form Γ1 J1 Γn ... 9) in which the global hypotheses have been made explicit. A hypothetical inductive definition is to be regarded as an ordinary inductive definition of a formal derivability judgement Γ J consisting of a finite set of basic judgements, Γ, and a basic judgement, J. A collection of hypothetical rules, R, defines the strongest formal derivability judgement that is structural and closed under rules R. 11c) These rules ensure that formal derivability behaves like a hypothetical judgement.

The principle of rule induction is simply the expression of the definition of an inductively defined judgement form as the strongest judgement form closed under the rules comprising the definition. This means that the judgement form defined by a set of rules is both (a) closed under those rules, and (b) sufficient for any other property also closed under those rules. The former means that a derivation is evidence for the validity of a judgement; the latter means that we may reason about an inductively defined judgement form by rule induction.

2). It follows directly from the definition of derivability that it is stable under extension with new rules. 1 (Stability). If Γ R J, then Γ R∪R J. Proof. Any derivation of J from R[Γ] is also a derivation from (R ∪ R )[Γ], since any rule in R is also a rule in R ∪ R . Derivability enjoys a number of structural properties that follow from its definition, independently of the rules, R, in question. Reflexivity Every judgement is a consequence of itself: Γ, J hypothesis justifies itself as conclusion.

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